Introduction to Probability and Statistics for Engineers and Scientists

4.3Jointly Distributed Random Variables 95

That is, the density is the derivative of the cumulative distribution function. A somewhat
more intuitive interpretation of the density function may be obtained from Equation 4.2.2
as follows:

P

{

a−

ε

2

≤X≤a+

ε

2

}

=

∫a+ε/2

a−ε/2

f(x)dx≈εf(a)

whenεis small. In other words, the probability thatXwill be contained in an interval
of lengthεaround the pointais approximatelyεf(a). From this, we see thatf(a)is
a measure of how likely it is that the random variable will be neara.

EXAMPLE 4.2b Suppose thatXis a continuous random variable whose probability density
function is given by

f(x)=

{

C(4x− 2 x^2 )0<x< 2

0 otherwise

(a)What is the value ofC?

(b)FindP{X> 1 }.

SOLUTION∫ (a) Since f is a probability density function, we must have that
∞
−∞f(x)dx=1, implying that

C

∫ 2

0

(4x− 2 x^2 )dx= 1

or

C

[

2 x^2 −

2 x^3

3

]∣

∣∣x=^2

x= 0

= 1

or

C=^38

(b)Hence

P{X> 1 }=

∫∞

1

f(x)dx=^38

∫ 2

1

(4x− 2 x^2 )dx=^12 ■

4.3Jointly Distributed Random Variables

For a given experiment, we are often interested not only in probability distribution
functions of individual random variables but also in the relationships between two or
more random variables. For instance, in an experiment into the possible causes of cancer,